Use square root and cube root symbols to represent solutions to equations of the form x<sup>2</sup> = p and x<sup>3</sup> = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that â2 is irrational.

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 Ã 10<sup>8</sup> and the population of the world as 7 Ã 10<sup>9</sup>, and determine that the world population is more than 20 times larger.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Start Up

15 minutes

The goal today is to make sure students get exposure to the typical variety of scientific notation word problems. However, the biggest struggle in solving most word problems is for students to get past the first step. If I can get students to believe in themselves and get past their trepidation and frustration surrounding word problems, then all of the strategies they have acquired during this unit will carry them forward.

I start this class, but asking students to write three words that come to mind when they see the phrase "word problems." I ask students to write these answers down and then we share as a class. Sometimes I ask students to to write the words on sticky notes and then put them on the board. The process is important. I like to get all of the negative energy out of the way (and believe me when I say that there is quite a bit of negative energy around word problems in my classes). We talk about why there is so much negative energy and what we can do to combat negative feelings. The short of it is that students tire of solving problems that are worded badly, feel artificial, and are presented in a way that would make the most avid problem solver nauseous.

My general teaching philosophy is aimed at redesigning the setbacks of these types of word problems, but for this lesson we take the standard type of word problem (something that students associate with torture and other unfortunate parts of life) and kick its butt. Earlier in the year, we discussed problem solving strategies by framing Polya's How to Solve it in words that work for students. Today, we put Polya's ideas to work.

Investigation

25 minutes

I hand out the Scientific Notation word problems (which is poorly presented and spaced on purpose) and ask kids what we can do to make these problems a little less overwhelming. If they are stuck, we can ask them what they don't like about the sheet. We then work our way toward one of our problem solving fundamentals: if a problem feels like too much, break it down.

I then encourage students to literally break this sheet up before they begin. I ask them to cut the sheet into strips and then put the papers into their binder (one per page). By separating the group, they can take on one problem at a time. By placing them on separate pages in their binder, they give themselves the whitespace needed to organize their thoughts.

During the process, I ask them to pick the problems they want to solve first (at least 3 of the group) and then finish the rest at home. There is no need to cram in all the problems in one sitting.

As the students read the problems and sort them into groups, I ask them why they chose certain problems. I ask them why they avoided the other problems. This is a chance to talk about common structure in problems and help students recognize what they are and aren't comfortable with.

Scientific Notation word problems.docx

Summary

20 minutes

Here the goal is to review a few problems and discuss strategies that helped them break the problem down. We usually are able to review about 4 problems and then students can review help material at home (I often provide help videos for students). The goal is to pick problems that highlight the benefit of certain problem solving strategies.

For example, I would highlight the important of drawing a diagram in problem 3:

Mercury has an average distance from the Sun is 57,910,000 km. The Earth is approximately 93,000,000 miles from the sun. How far is a trip from Mercury to the Sun and then back to Earth? Write your result in scientific notation.

Here the problem becomes much easier when you draw three points for Mercury, the Sun and Earth. Once the diagram is drawn, you realize all you need to do is add the distances.

Problem 9 relates scale and area, something that can be drawn to better understand:

You have a field that is equal to 314 square kilometers. How many square meters are in the field? 1 square kilometer = 1 million square meters. Express your answer is scientific notation.

Problem 8 is a beast (meaning a real tough problem) for middle school students:

If the mass of the Earth is approximately 5.974 x 1024kilograms and the radius of the Earth is approximately 6.3 x 106 meters, what is density of the Earth? Assume the Earth is spherical.

The problem is difficult because students need to understand that they must find the volume of the Earth and then make their calculation. By sketching the Earth and the given radius, they can organize their approach.

Its not that they need to draw to solve any of these problems, its just that this is a strategy that could really help. I always invite students to share their own approach.

I extend the conversation by asking questions like, "which problems were really asking you to just multiply or divide? How did you know?" The goal is ask students what strategies they used and which problems lent themselves to that strategy.